Efficient SAV Algorithms for Curvature Minimization Problems

The curvature regularization method is well-known for its good geometric interpretability and strong priors in the continuity of edges, which has been applied to various image processing tasks. However, due to the non-convex, non-smooth, and highly non-linear intrinsic limitations, most existing algorithms lack a convergence guarantee. This paper proposes an efficient yet accurate scalar auxiliary variable (SAV) scheme for solving both mean curvature and Gaussian curvature minimization problems. The SAV-based algorithms are shown unconditionally energy diminishing, fast convergent, and very easy to be implemented for different image applications. Numerical experiments on noise removal, image deblurring, and single image super-resolution are presented on both gray and color image datasets to demonstrate the robustness and efficiency of our method. Source codes are made publicly available at https://github.com/Duanlab123/SAV-curvature.

[1]  Jiantao Zhou,et al.  Multistage Curvature-Guided Network for Progressive Single Image Reflection Removal , 2022, IEEE Transactions on Circuits and Systems for Video Technology.

[2]  T. Zeng,et al.  Efficient Boosted DC Algorithm for Nonconvex Image Restoration with Rician Noise , 2022, SIAM J. Imaging Sci..

[3]  Guojia Hou,et al.  A Variational Framework for Underwater Image Dehazing and Deblurring , 2021, IEEE Transactions on Circuits and Systems for Video Technology.

[4]  R. Glowinski,et al.  An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications in Surface Smoothing and Imaging , 2021, SIAM J. Sci. Comput..

[5]  T. Zeng,et al.  Edge adaptive hybrid regularization model for image deblurring , 2020, Inverse Problems.

[6]  Yushun Wang,et al.  High-order structure-preserving algorithms for the multi-dimensional fractional nonlinear Schrödinger equation based on the SAV approach , 2021, Math. Comput. Simul..

[7]  Z. Wen,et al.  Enhance Curvature Information by Structured Stochastic Quasi-Newton Methods , 2021, 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[8]  Jie Shen,et al.  Scalar Auxiliary Variable/Lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations , 2021, J. Comput. Phys..

[9]  Xue-Cheng Tai,et al.  Efficient and Convergent Preconditioned ADMM for the Potts Models , 2021, SIAM J. Sci. Comput..

[10]  Junxiang Yang,et al.  An improved scalar auxiliary variable (SAV) approach for the phase-field surfactant model , 2021 .

[11]  Yuping Duan,et al.  Image Reconstruction by Minimizing Curvatures on Image Surface , 2020, J. Math. Imaging Vis..

[12]  Yue M. Lu,et al.  Bilinear Constraint based ADMM for Mixed Poisson-Gaussian Noise Removal , 2019, Inverse Problems & Imaging.

[13]  Zuoqiang Shi,et al.  An Unsupervised Deep Learning Approach for Real-World Image Denoising , 2021, International Conference on Learning Representations.

[14]  Kevin Chen-Chuan Chang,et al.  Curvature Regularization to Prevent Distortion in Graph Embedding , 2020, NeurIPS.

[15]  Yuping Duan,et al.  Minimizing Discrete Total Curvature for Image Processing , 2020, 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[16]  Wenjuan Yao,et al.  A Total Fractional-Order Variation Model for Image Super-Resolution and Its SAV Algorithm , 2020, J. Sci. Comput..

[17]  Xiaoli Li,et al.  The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing , 2019, SIAM J. Sci. Comput..

[18]  Jie Shen,et al.  Efficient SAV approach for imaginary time gradient flows with applications to one- and multi-component Bose-Einstein Condensates , 2019, J. Comput. Phys..

[19]  Yuanhao Gong,et al.  Mean Curvature Is a Good Regularization for Image Processing , 2019, IEEE Transactions on Circuits and Systems for Video Technology.

[20]  Seyed-Mohsen Moosavi-Dezfooli,et al.  Robustness via Curvature Regularization, and Vice Versa , 2018, 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[21]  Wangmeng Zuo,et al.  Toward Convolutional Blind Denoising of Real Photographs , 2018, 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[22]  Suchuan Dong,et al.  Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable , 2018, J. Comput. Phys..

[23]  Jie Shen,et al.  A New Class of Efficient and Robust Energy Stable Schemes for Gradient Flows , 2017, SIAM Rev..

[24]  Wotao Yin,et al.  Global Convergence of ADMM in Nonconvex Nonsmooth Optimization , 2015, Journal of Scientific Computing.

[25]  Jiang Yang,et al.  The scalar auxiliary variable (SAV) approach for gradient flows , 2018, J. Comput. Phys..

[26]  Ming-Hsuan Yang,et al.  $L_0$ -Regularized Intensity and Gradient Prior for Deblurring Text Images and Beyond , 2017, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[27]  Carola-Bibiane Schönlieb,et al.  Infimal Convolution of Data Discrepancies for Mixed Noise Removal , 2016, SIAM J. Imaging Sci..

[28]  Lei Zhang,et al.  Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising , 2016, IEEE Transactions on Image Processing.

[29]  Xiaofeng Yang,et al.  Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends , 2016, J. Comput. Phys..

[30]  Ke Chen,et al.  Image denoising using the Gaussian curvature of the image surface , 2016 .

[31]  A. Basarab,et al.  Fast Single Image Super-resolution using a New Analytical Solution for l2-l2 Problems. , 2016, IEEE transactions on image processing : a publication of the IEEE Signal Processing Society.

[32]  Maria Drangova,et al.  Thin Structure Estimation with Curvature Regularization , 2015, 2015 IEEE International Conference on Computer Vision (ICCV).

[33]  Zhi-Quan Luo,et al.  Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems , 2014, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[34]  Xue-Cheng Tai,et al.  Augmented Lagrangian method for a mean curvature based image denoising model , 2013 .

[35]  Carola-Bibiane Schönlieb,et al.  A Combined First and Second Order Variational Approach for Image Reconstruction , 2012, Journal of Mathematical Imaging and Vision.

[36]  Tony F. Chan,et al.  Image Denoising Using Mean Curvature of Image Surface , 2012, SIAM J. Imaging Sci..

[37]  Lei Zhang,et al.  Color demosaicking by local directional interpolation and nonlocal adaptive thresholding , 2011, J. Electronic Imaging.

[38]  Xue-Cheng Tai,et al.  A Fast Algorithm for Euler's Elastica Model Using Augmented Lagrangian Method , 2011, SIAM J. Imaging Sci..

[39]  Xue-Cheng Tai,et al.  A Fast Algorithm for a Mean Curvature Based Image Denoising Model Using Augmented Lagrangian Method , 2011, Efficient Algorithms for Global Optimization Methods in Computer Vision.

[40]  Ke Chen,et al.  Multigrid Algorithm for High Order Denoising , 2010, SIAM J. Imaging Sci..

[41]  Michael Elad,et al.  On Single Image Scale-Up Using Sparse-Representations , 2010, Curves and Surfaces.

[42]  Xiaofeng Yang,et al.  Numerical approximations of Allen-Cahn and Cahn-Hilliard equations , 2010 .

[43]  Michael J. Black,et al.  Fields of Experts , 2009, International Journal of Computer Vision.

[44]  Alessandro Foi,et al.  Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering , 2007, IEEE Transactions on Image Processing.

[45]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .